Overview
This lecture covers kinematic problem-solving, focusing on displacement and distance in constant acceleration motion, the difference between them, interpreting velocity-time (VT) graphs, and introduces a classic "chasing problem." The session ends with an introduction to vertical motion under gravity.
Displacement vs. Distance (Problem 1.9, Parts C & D)
- Displacement is calculated as x_final minus x_initial, independent of the path taken.
- Turning points (where velocity momentarily becomes zero) do not impact displacement, only initial and final positions do.
- Constant acceleration allows direct use of the kinematic equation: ( v_f^2 = v_i^2 + 2a\Delta x ).
- For distance, sum the absolute values of each segment’s displacement, including before and after any turning point.
- Use separated calculations for each segment (e.g., ( \Delta x_1 ) and ( \Delta x_2 )), then take |Δx₁| + |Δx₂| to find total distance.
- On VT graphs, displacement is the area under the curve (can be positive or negative); distance is the sum of absolute areas.
Velocity-Time (VT) Graph Interpretation
- The slope of a VT graph gives acceleration (constant if the line is straight).
- Negative areas on a VT graph indicate motion in the negative direction.
- The sum of signed areas under the VT curve equals total displacement.
Chasing Problem (Homework Problem 10)
- Involves two persons: Person A (starts from rest with acceleration) and Person B (moves at constant speed, starts earlier).
- Both positions are expressed as functions of time using kinematic equations.
- The meeting point is found by setting Person A’s and Person B’s positions equal and solving a quadratic equation for time.
- Only positive roots of the quadratic equation are physically meaningful.
- Distance traveled by Person A is calculated using ( x = \frac{1}{2}at^2 ) with the found time.
Intro to Vertical (Projectile) Motion
- Vertical motion under gravity is treated with the same kinematic equations, but acceleration ( a_y = -9.8\ m/s^2 ) (downward).
- Throwing a ball upward: initial velocity is upward, gravity opposes the motion.
- At the highest point, vertical velocity is zero (a turning point).
- Four kinematic equations apply, with variables adapted for vertical (y) direction.
Key Terms & Definitions
- Displacement — Change in position: final position minus initial position.
- Distance — Total length traveled, regardless of direction; sum of absolute displacements.
- Turning Point — Momentary stop where velocity is zero before reversing direction.
- Kinematic Equations — Equations describing motion with constant acceleration.
- VT Graph — Graph showing velocity as a function of time.
- Projectile Motion — Motion of an object thrown or projected, subject only to gravity.
Action Items / Next Steps
- Complete homework questions 1–11 before moving to section 8.
- Review examples in the textbook related to section 7.
- Watch the recorded video lecture to review the four kinematic equations.
- Prepare for new problems involving vertical (projectile) motion in the next lecture.